
theorem Th35:
  for T being non empty TopSpace
  for x,y being Element of InclPoset the topology of T
  st for F being Subset-Family of T st F is open & y c= union F
  ex G being finite Subset of F st x c= union G holds x is_way_below y
proof
  let T be non empty TopSpace;
  set L = InclPoset the topology of T;
A1: L = RelStr(#the topology of T, RelIncl the topology of T#)
  by YELLOW_1:def 1;
  let x,y be Element of L such that
A2: for F being Subset-Family of T st F is open & y c= union F
  ex G being finite Subset of F st x c= union G;
  now
    let I be Ideal of L;
    reconsider F = I as Subset-Family of T by A1,XBOOLE_1:1;
    assume y <= sup I;
    then y c= sup I by YELLOW_1:3;
    then
A3: y c= union F by YELLOW_1:22;
    F is open by YELLOW_1:25;
    then consider G being finite Subset of F such that
A4: x c= union G by A2,A3;
    reconsider G as finite Subset of L by XBOOLE_1:1;
    consider z being Element of L such that
A5: z in I and
A6: z is_>=_than G by WAYBEL_0:1;
A7: union G = sup G by YELLOW_1:22;
A8: z >= sup G by A6,YELLOW_0:32;
A9: x <= sup G by A4,A7,YELLOW_1:3;
    sup G in I by A5,A8,WAYBEL_0:def 19;
    hence x in I by A9,WAYBEL_0:def 19;
  end;
  hence thesis by Th21;
end;
